Model Predictive Control based on LMIs Applied to an

Model Predictive Control based on LMIs Applied to an Omni-Directional Mobile Robot Humberto X. Araújo ∗ André G. S. Conceição ∗ Gustavo H. C. Oliveira ∗∗ Jônatas Pitanga ∗ ∗

Electrical Engineering Department (DEE), Universidade Federal da Bahia (UFBA), e-mail: [email protected], [email protected], [email protected] ∗∗ Electrical Engineering Department (DELT), Universidade Federal do Paraná (UFPR), e-mail: [email protected] Abstract: The paper presents a methodology for state feedback MPC synthesis applied to the trajectory tracking control problem of a three wheeled omnidirectional mobile robot. The MPC design used here is based on a cost function developed over finite horizon and LMI framework. It is shown that MPC concepts well established for robot applications, for instance, the use of open loop predictions, receding horizon control, constraints manipulation, are preserved in this new formulation. The stability of the closed loop system is guaranteed by LMI conditions related with the cost function monotonicity. Simulation results of navigation are provided to demonstrate the performance of the proposed control strategy. Keywords: Omni-Directional Robot, Model Predictive Control, Trajectory Control, Closed-loop Stability, Input constraints, Linear Matrix Inequalities. 1. INTRODUCTION In the last years, there has been a great interest in model based predictive control techniques for trajectory tracking of mobile robots, as it can be seen in Oliveira and Carvalho [2003], Kanjanawanishkul and Zell [2008], Li and Zell [2008] and Raffo et al. [2009], for obstacle avoidance, as it is presented in Lim et al. [2008] and Shim et al. [2006], and applied to diverse locomotion mechanisms in Anderson and Stone [2007], Lim et al. [2009], Kang and Hedrick [2009]. Many methods in this context have been developed. In Jiang et al. [2005], a tracking method for a mobile robot is presented, where the predictive control is used to predict the position and the orientation of the robot, while the fuzzy control is used to deal with the nonlinear characteristics of the system. A path tracking scheme for mobile robot based on neural predictive control is presented in Gu and Hu [2002], where a multi-layer back-propagation neural network is employed to model non-linear kinematics of the robot. In Ramirez et al. [1999], genetic algorithms are used to perform online nonlinear optimization of the cost function. In this work, the approach to perform the minimization of the controller’s cost function is based on convex optimization, involving LMIs. Many reasons justify the use of LMIs (Boyd et al. [1994]). LMI-based optimization problems can be solved in polynomial time, using interior-point methods and the optimal solution is global. Omnidirectional mobile robots have the ability to move simultaneously and independently in translation and rota-

tion Pin and Killough [1994]. However, nonlinearities, like motor dynamic constraints or friction related to robot’s velocity, can greatly affect the behavior of the robot. So, the predictive control technique must be able to take into account the nonlinearities. A quite simple model has be used in Conceição et al. [2009], where methods for parameter estimation were proposed and validated through experimental results. A similar model is used in this work. On the other hand, in the last decades, several LMI-based approaches have been proposed for design control laws, mainly in ℋ2 /ℋ∞ control. In the context of MPC, the use of LMIs was firstly proposed in the nineties, with the paper of Kothare et al. [1996]. In this work, a state space structure with parameter uncertainties is assumed and LMI-based constraints obtained from Lyapunov functions guarantee closed loop stability. These constraints should be satisfied for all model realizations. The state feedback control law minimizes an infinite horizon cost function subject to input and output constraints. Other references related with this approach are Mao [2003], Granado et al. [2003], Ding. et al. [2004], Casavola et al. [2004] and Lee and Park [2006]. The present paper contains a methodology for state feedback MPC synthesis based on the LMI framework, proposed in (Araujo and Oliveira [2011]), but here modified to approach the omnidirectional mobile robot case. An important property of the proposed algorithm is the use of finite horizon together with a sufficient condition for closed-loop stability. Constraints in the input and output signals are also tackled by the control law design. The stability condition is obtained by guaranteeing that an upper

bound for the finite horizon cost function is monotonically non-increasing. These approach for achieving closed loop stability is different for the one described in Kothare et al. [1996] and related works, for infinite horizon, where stability is achieved by the state feedback control law u = Fx (that is, a fixed structure for the control law) due to the well known change of variables 𝐹 = 𝑌 𝑄−1 . The controller architecture considers both the kinematic and the dynamic control in a cascade structure, where a MPC is used to make the robot follow a reference velocity trajectory, generated by an inverse kinematics block which translates position objectives into velocity references. The MPC strategy is characterized by the use of a process model, where control actions are calculated by optimization of a cost function. Taking into account this strategy, the problem of driving a mobile robot to follow a previously defined path can be applied to MPC. The future reference (desired path) is known a priori, so the robot can react before the change has effectively been made, avoiding effects of delay. The structure of this paper is presented in the following. In Section 2, the omnidirectional mobile robot is modeled into discrete space-state equations. In Section 3, the optimization problem is formulated. Section 4 contains the new synthesis of contrained stable MPC based on LMI framework. The trajectory generation method is defined in 5. Simulation results are presented in Section 6 and, the conclusions are addressed in Section 7. 2. THE OMNI-DIRECTIONAL MOBILE ROBOT MODEL The robot pose is fully described by the vector: 𝜉 = [𝑥𝑟 𝑦𝑟 𝜃]𝑇 , where 𝑥𝑟 and 𝑦𝑟 is the localization of the point 𝑃 in the world frame, and 𝜃 the angular difference between the world and robot frames. 𝜉˙𝑟 = [𝑣 𝑣𝑛 𝑤]𝑇 is the velocity vector at the robot frame and describes the linear velocity of the robot at the point 𝑃 , represented by the orthogonal components 𝑣 and 𝑣𝑛 , and the angular velocity of the robot’s body, represented by 𝑤. By geometric parameters of the robot and the robot’s body frame, in Fig. 1(b), the motion equations are as follows: 𝜉˙ = 𝑅𝑇 (𝜃)𝜉˙𝑟 (1) where an orthogonal rotation matrix 𝑅(𝜃) is defined to map the world frame into the robot frame, and vice versa: [ 𝑅(𝜃) =

𝑐𝑜𝑠(𝜃) 𝑠𝑖𝑛(𝜃) 0 −𝑠𝑖𝑛(𝜃) 𝑐𝑜𝑠(𝜃) 0 0 0 1

] (2)

(b) Geometric parameters and reference frames.

Fig. 1. Omnidirectional robot 𝑑𝑣𝑛 (𝑡) , 𝑑𝑡 𝑑𝑤(𝑡) Γ(𝑡) − 𝐵𝑤 𝑤(𝑡) − 𝐶𝑤 𝑠𝑔𝑛(𝑤(𝑡)) = 𝐼𝑛 , 𝑑𝑡

𝐹𝑣𝑛 (𝑡) − 𝐵𝑣𝑛 𝑣𝑛 (𝑡) − 𝐶𝑣𝑛 𝑠𝑔𝑛(𝑣𝑛 (𝑡)) = 𝑀

where,

{ 𝑠𝑔𝑛(𝛼) =

(4) (5)

1, 𝛼 > 0, 0, 𝛼 = 0, −1, 𝛼 < 0.

where 𝐹 = [𝐹𝑣 𝐹𝑣𝑛 Γ]𝑇 represents the vector force (𝐹𝑣 and 𝐹𝑣𝑛 ) in the robot frame and the moment (Γ) around the center of gravity for the mobile robot (point 𝑃 ). The robot mass is 𝑀 and the robot inertia moment is 𝐼𝑛 . The viscous frictions (𝐵𝑣 𝑣(𝑡), 𝐵𝑣𝑛 𝑣𝑛 (𝑡) and 𝐵𝑤 𝑤(𝑡)) represent a retarding force, which is a linear relationship between the applied force and the velocity. The coulomb frictions (𝐶𝑣 𝑠𝑔𝑛(𝑣(𝑡)), 𝐶𝑣𝑛 𝑠𝑔𝑛(𝑣𝑛 (𝑡)) and 𝐶𝑤 𝑠𝑔𝑛(𝑤(𝑡))) represent a retarding force that has constant amplitude with respect to the change of velocity, but the sign changes with the reversal of the direction of velocity, Kuo [1995]. 𝐵𝑣 , 𝐵𝑣𝑛 and 𝐵𝑤 are the viscous friction coefficients related to 𝑣, 𝑣𝑛 and 𝑤 respectively, and 𝐶𝑣 , 𝐶𝑣𝑛 and 𝐶𝑤 the coulomb friction coefficients. The relationships between the robot’s traction forces and the wheel’s traction forces are, 𝐹𝑣 (𝑡) = 𝑓2 (𝑡)𝑐𝑜𝑠(𝛿) − 𝑓3 (𝑡)𝑐𝑜𝑠(𝛿), 𝐹𝑣𝑛 (𝑡) = −𝑓1 (𝑡) + 𝑓2 (𝑡)𝑠𝑖𝑛(𝛿) + 𝑓3 (𝑡)𝑠𝑖𝑛(𝛿),

(6) (7)

Γ(𝑡) = (𝑓1 (𝑡) + 𝑓2 (𝑡) + 𝑓3 (𝑡))𝑏. (8) where 𝛿 is 𝜋/6. The wheel’s traction force on each wheel 𝑖 (for 𝑖 = 1, ...3) is 𝑇𝑖 (𝑡) 𝑓𝑖 (𝑡) = . (9) 𝑟𝑖 with 𝑇𝑖 the rotation torque of the wheels.

The model is developed based on the dynamics, kinematics and dynamics of DC motors of the robot. This prediction model could find nonlinear saturation elements, such as limitations of current of the motors, amplitude of the applied voltage in the motors, and frictions related to the equation of motion. By Newton’s law of motion and the robot’s body frame, in Fig. 1(b), the dynamic model is described by 𝐹𝑣 (𝑡) − 𝐵𝑣 𝑣(𝑡) − 𝐶𝑣 𝑠𝑔𝑛(𝑣(𝑡)) = 𝑀

(a) Mobile robot.

𝑑𝑣(𝑡) , 𝑑𝑡

(3)

The dynamics of each DC motor 𝑖 (for 𝑖 = 1, ..., 3) can be described using the following equations, 𝑑𝑖𝑎𝑖 (𝑡) + 𝑅𝑎𝑖 𝑖𝑎𝑖 (𝑡) + 𝐾𝑣𝑖 𝑤𝑚𝑖 (𝑡), (10) 𝑑𝑡 𝑇𝑖 (𝑡) = 𝑙𝑖 𝐾𝑡𝑖 𝑖𝑎𝑖 (𝑡), 0 ≤ 𝑖𝑎𝑖 (𝑡) ≤ 𝑖𝑚𝑎𝑥 (11) where 𝐿𝑎𝑖 are the motor’s armature inductance, 𝑅𝑎𝑖 are the motor’s armature resistance, 𝑙𝑖 are the motor’s gear ratio reduction, and 𝑤𝑚𝑖 the angular velocity of the wheels. The motor electric circuit dynamics can be neglected , 𝑢𝑖 (𝑡) = 𝐿𝑎𝑖

𝑑𝑖

(𝑡)

which leads to 𝐿𝑎𝑖 𝑎𝑑𝑡𝑖 = 0. In summary, motor’s parameters, geometric parameters, and estimated dynamic parameters are shown in Table 1.

𝐽(𝑘) =

Description wheel radius motor reduction robot inertia moment robot mass viscous friction related to 𝑣 viscous friction related to 𝑣𝑛 viscous friction related to 𝑤 coulomb friction related to 𝑣 coulomb friction related to 𝑣𝑛 coulomb friction related to 𝑤 motor’s armature resistance motor’s emf constant distance between P and wheels

Value 0.035 19 0.05 1.5 1.79 1.82 0.016 0.82 0.58 0.092 1.71 0.0059 0.1

3. LMI FORMULATION OF FINITE HORIZON CONSTRAINED MPC FOR A CLASS OF MOBILE ROBOTS The state-space equations in discrete-time obtained for the mobile robot are given by: { 𝑥(𝑘 + 1) = 𝐴𝑥(𝑘) + 𝐵𝑢(𝑘) + 𝐾𝑠𝑖𝑔𝑛(𝑥(𝑘)) (12) 𝑦(𝑘) = 𝐶𝑥(𝑘), where 𝑢(𝑘) = [𝑢1 (𝑘) 𝑢2 (𝑘) 𝑢3 (𝑘)]𝑇 ∈ ℛ𝑛𝑢 , 𝑛𝑢 = 3, is the control input corresponding to the motor’s armature voltages, the output and the state-variables of the system are the robot velocities, 𝑦(𝑘) = 𝑥(𝑘) = [𝑣(𝑘) 𝑣𝑛 (𝑘) 𝑤(𝑘)]𝑇 ∈ ℛ𝑛𝑦 , 𝑛𝑦 = 3. The state-variable-form matrices defining this system are −1 𝑇 𝐴 = −𝐺𝑄𝐾𝑡 𝐿𝑅𝑀 𝐺 − 𝐾1

(13)

𝐵 = 𝐺𝑄

(14)

⎡ 𝐶𝑣 0 − ⎢ 𝑀 𝐶𝑣𝑛 𝐾=⎢ 0 − ⎣ 𝑀 0

0

⎤

⎡ 𝐵𝑣 0 ⎥ ⎢ 𝑀 𝐵𝑣𝑛 ⎥ , 𝐾1 = ⎢ 0 ⎦ ⎣ 𝑀

0 0 𝐶𝑤 − 𝐼𝑛

0

⎡ 𝑙 𝐾 1 𝑡1 0 ⎢ 𝑀 𝑅𝑎1 𝑟1 𝑙2 𝐾𝑡2 ⎢ 𝑄=⎢ 0 𝑀 𝑅𝑎2 𝑟2 ⎣ 0

[ 𝐺=

[ 𝐶=

1 0 0 0 1 0 0 0 1

0

]

]

[

,

𝑅𝑀 =

𝑙3 𝐾𝑡3 𝑀 𝑅𝑎3 𝑟3 ,

[

𝐾𝑡 =

𝑟1 0 0 0 𝑟2 0 0 0 𝑟3

0 𝐵𝑤 𝐼𝑛

⎤

⎥ ⎥ , (15) ⎦

⎥ ⎥, ⎥ ⎦

0

0

0 𝑐𝑜𝑠(𝛿) −𝑐𝑜𝑠(𝛿) −1 𝑠𝑖𝑛(𝛿) 𝑠𝑖𝑛(𝛿) 𝑏 𝑏 𝑏

0

⎤

0

]

(16)

+ ∥Δ𝑢(𝑘 + 𝑗 −

𝐾𝑡1 0 0 0 𝐾𝑡2 0 0 0 𝐾𝑡3

[

,

𝐿=

𝑙1 0 0 0 𝑙2 0 0 0 𝑙3

,

(17)

2 1∣𝑘)∥𝑅𝑗−1

,

The control law is obtained by minimizing cost function (20) in relation to the control moves, that is: min 𝐽(𝑘), (21) Δ𝑢(𝑘∣𝑘)⋅⋅⋅Δ𝑢(𝑘+𝑁𝑢 −1∣𝑘)

where 𝑁𝑢 is the control horizon. It is assumed that the control input is constant after sample time 𝑘 + 𝑁𝑢 − 1, i.e., Δ𝑢(𝑘 + 𝑗∣𝑘) = 0 for 𝑗 ≥ 𝑁𝑢 . At each sampling time 𝑘, the vector Δu containing the sequence of control moves is calculated. Only the first control move is applied to the system. Defining the vectors ⎡ ⎤ ⎡ ⎤ 𝑦ˆ(𝑘 + 1∣𝑘) Δ𝑢(𝑘∣𝑘) ⎢ 𝑦ˆ(𝑘 + 2∣𝑘) ⎥ ⎢ Δ𝑢(𝑘 + 1∣𝑘) ⎥ ⎥ Δu = ⎢ ⎥ , (22) y ˆ=⎢ .. .. ⎣ ⎦ ⎣ ⎦ . . 𝑦ˆ(𝑘 + 𝑁𝑦 ∣𝑘) Δ𝑢(𝑘 + 𝑁𝑢 − 1∣𝑘) the predicted outputs can be calculated over the finite horizon by y ˆ = 𝐻Δu + 𝐹 Δ𝑥(𝑘) + Π𝑦(𝑘), (23) where

⎡

⎤ 𝐶𝑆0 𝐵 0 0 ⋅⋅⋅ 𝐶𝑆0 𝐵 0 ⋅⋅⋅⎥ ⎢ 𝐶𝑆1 𝐵 ⎥, 𝐻=⎢ .. .. .. ⎣ ⎦ . . . 𝐶𝑆𝑁𝑦 −1 𝐵 𝐶𝑆𝑁𝑦 −2 𝐵 𝐶𝑆𝑁𝑦 −3 ⋅ ⋅ ⋅ ⎡ ⎤ 𝐶𝑆0 𝐴 ⎢ 𝐶𝑆1 𝐴 ⎥ 𝑇 ⎥, 𝐹 =⎢ Π = [𝐼 𝐼 𝐼 ⋅⋅⋅ 𝐼 ] , .. ⎣ ⎦ . 𝐶𝑆𝑁𝑦 −1 𝐴 ∑𝑗 𝑖=0

𝐴𝑖 .

Hence, the objective function in optimization problem (21) is equivalent to

]

𝑇

𝐽(𝑘) = (ˆ y − r) 𝑄𝑦 (ˆ y − r) + Δu𝑇 𝑅Δu, . (18)

Defining Δ𝑢(𝑘) = 𝑢(𝑘) − 𝑢(𝑘 − 1) and Δ𝑥(𝑘) = 𝑥(𝑘) − 𝑥(𝑘 − 1), system (12) can be rewritten by: { Δ𝑥(𝑘 + 1) = 𝐴Δ𝑥(𝑘) + 𝐵Δ𝑢(𝑘)+ +𝐾[𝑠𝑖𝑔𝑛(𝑥(𝑘)) − 𝑠𝑖𝑔𝑛(𝑥(𝑘 − 1))] (19) 𝑦(𝑘 + 1) = 𝐶Δ𝑥(𝑘 + 1) + 𝑦(𝑘). In this work, a quadratic objective function over finite horizon, for the first 𝑁𝑦 steps, is given by:

(20)

where 𝑁𝑦 defines the prediction horizon, 𝑄𝑗 ≥ 0 and 𝑅𝑗 > 0 are suitable weighting matrices, 𝑦ˆ(𝑘 + 𝑗∣𝑘) and Δ𝑢(𝑘 + 𝑗 − 1∣𝑘) are the predicted output at 𝑘 + 𝑗 and the optimal value for the incremental control at 𝑘 + 𝑗 − 1, respectively, computed at time 𝑘, by using the model composed by equations (19), and 𝑟(𝑘 + 𝑗) is the set-point at time 𝑘 + 𝑗.

and 𝑆𝑗 =

]

2

∥ˆ 𝑦 (𝑘 + 𝑗∣𝑘) − 𝑟(𝑘 + 𝑗)∥𝑄𝑗

𝑗=1

Table 1. Parameters of the model. Symbol 𝑟1 , 𝑟2 , 𝑟3 (𝑚) 𝑙1 , 𝑙2 , 𝑙3 𝐼𝑛 (𝑘𝑔.𝑚2 ) 𝑀 (𝐾𝑔) 𝐵𝑣 (𝑁/𝑚/𝑠) 𝐵𝑣𝑛 (𝑁/𝑚/𝑠) 𝐵𝑤 (𝑁.𝑚/𝑟𝑎𝑑/𝑠) 𝐶𝑣 (𝑁 ) 𝐶𝑣𝑛 (𝑁 ) 𝐶𝑤 (𝑁.𝑚) 𝑅𝑎1..3 (Ω) 𝐾𝑡1..3 (𝑉 𝑜𝑙𝑡𝑠/𝑟𝑎𝑑/𝑠) 𝑏(𝑚)

𝑁𝑦 ∑

where 𝑄𝑦 = diag(𝑄1 , 𝑄2 , ⋅ ⋅ ⋅ , 𝑄𝑁𝑦 ), 𝑅 = diag(𝑅0 , 𝑅1 , ⋅ ⋅ ⋅ , 𝑅𝑁𝑢 −1 ), and r is the reference input given by ⎡ ⎤ 𝑟(𝑘 + 1) ⎢ 𝑟(𝑘 + 2) ⎥ ⎥, r=⎢ .. ⎣ ⎦ . 𝑟(𝑘 + 𝑁𝑦 )

(24)

with y ˆ given by (23). The system state is assumed available at each sampling time 𝑘, by measurement or estimation. In this work, constraints on actuators are taken in account by component-wise peak bounds on the input 𝑢(𝑘 + 𝑗∣𝑘) and its variation: 𝑢𝑚𝑖𝑛𝑗𝑖 ≤ 𝑢𝑖 (𝑘 + 𝑗∣𝑘) ≤ 𝑢𝑚𝑎𝑥𝑗𝑖 , (25) 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛𝑢 ; 𝑘, 𝑗 ≥ 0, 𝑣𝑚𝑖𝑛𝑗𝑖 ≤ Δ𝑢𝑖 (𝑘 + 𝑗∣𝑘) ≤ 𝑣𝑚𝑎𝑥𝑗𝑖 , (26) 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛𝑢 ; 𝑘, 𝑗 ≥ 0, where 𝑢𝑖 (𝑘) and Δ𝑢𝑖 (𝑘 + 𝑗∣𝑘) represent the 𝑖-th input in the vectors 𝑢(𝑘) and Δ𝑢(𝑘 + 𝑗∣𝑘), respectively. Similarly, for the output, representing performance requirements, component-wise peak bounds on 𝑦(𝑘 + 𝑗) can also be considered: 𝑦𝑚𝑖𝑛𝑗𝑖 ≤ 𝑦𝑖 (𝑘 + 𝑗) ≤ 𝑦𝑚𝑎𝑥𝑗𝑖 , (27) 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛𝑦 ; 𝑘 ≥ 0, 𝑗 ≥ 1. In this work, an LMI-based synthesis procedure is provided for finite horizon MPC optimization problem (21), with input and output constraints. Next, the representation of the input and output constraints in terms of LMI is presented. 3.1 Input constraints Consider component-wise peak bounds on input (25) at each present and future sampling time 𝑘: 𝑢𝑚𝑖𝑛𝑗𝑖 ≤ 𝑢𝑖 (𝑘 + 𝑗∣𝑘) ≤ 𝑢𝑚𝑎𝑥𝑗𝑖 , (28) 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛𝑢 ; 𝑘 ≥ 0, 𝑗 = 0, 1, ⋅ ⋅ ⋅ , 𝑁𝑢 − 1. Since ⎧ = Δ𝑢(𝑘∣𝑘) + 𝑢(𝑘 − 1)  ⎨ 𝑢(𝑘∣𝑘) 𝑢(𝑘 + 1∣𝑘) = Δ𝑢(𝑘 + 1∣𝑘) + Δ𝑢(𝑘∣𝑘) + 𝑢(𝑘 − 1) (29) ..  ⎩ . inequalities (28) can be rearranged, using Schur complement, as the set of 𝑛𝑢 𝑁𝑢 inequalities on variable Δu: 𝑢𝑚𝑖𝑛0𝑖 − 𝑢𝑖 (𝑘 − 1) ≤ ([ 𝐼 0 0 ⋅ ⋅ ⋅ 0 ])𝑖 Δu ≤ 𝑢𝑚𝑎𝑥0𝑖 − 𝑢𝑖 (𝑘 − 1), 𝑢𝑚𝑖𝑛1𝑖 − 𝑢𝑖 (𝑘 − 1) ≤ ([ 𝐼 𝐼 0 ⋅ ⋅ ⋅ 0 ])𝑖 Δu ≤ 𝑢𝑚𝑎𝑥1𝑖 − 𝑢𝑖 (𝑘 − 1), .. . 𝑢𝑚𝑖𝑛(𝑁𝑢 −1)𝑖 − 𝑢𝑖 (𝑘 − 1) ≤ ([ 𝐼 𝐼 𝐼 ⋅ ⋅ ⋅ 𝐼 ])𝑖 Δu ≤ 𝑢𝑚𝑎𝑥(𝑁𝑢 −1)𝑖 − 𝑢𝑖 (𝑘 − 1),

(30)

with 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛𝑢 . The entry 𝑢(𝑘 − 1) corresponds to the control move implemented at the last iteration. The notation (𝑀 )𝑖 means that only the row 𝑖 of matrix M is taken. Each side of inequalities (30) are LMI in Δu. Hence, there is a set of 2𝑛𝑢 𝑁𝑢 LMIs. To guarantee constraints (26), the following inequalities can be determined from (29) 𝑣𝑚𝑖𝑛0𝑖 ≤ ([ 𝐼 0 0 ⋅ ⋅ ⋅ 0 ])𝑖 Δu ≤ 𝑣𝑚𝑎𝑥0𝑖 , 𝑣𝑚𝑖𝑛1𝑖 ≤ ([ 0 𝐼 0 ⋅ ⋅ ⋅ 0 ])𝑖 Δu ≤ 𝑣𝑚𝑎𝑥1𝑖 ,

𝑣𝑚𝑖𝑛(𝑁𝑢 −1)𝑖

.. . 0 0 0 ⋅ ⋅ ⋅ 𝐼 ≤ ([ ])𝑖 Δu ≤ 𝑣𝑚𝑎𝑥(𝑁𝑢 −1)𝑖 ,

(31)

with 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛𝑢 . Inequalities (31) are a set of 2𝑛𝑢 𝑁𝑢 LMIs in Δ𝑢. LMIs (30) and (31) represent necessary and sufficient conditions to impose constraints on the input. 3.2 Output constraints Consider component-wise peak bounds on output (27) at each future sampling time 𝑘: 𝑦𝑚𝑖𝑛𝑗𝑖 ≤ 𝑦ˆ𝑖 (𝑘 + 𝑗∣𝑘) ≤ 𝑦𝑚𝑎𝑥𝑗𝑖 , 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛𝑦 ; 𝑘 ≥ 0, 𝑗 ≥ 1. Since, at sampling time 𝑘, ( ) ( ) 𝑦𝑚𝑖𝑛𝑗𝑖 ≤ 𝐻 (𝑗) Δu + 𝐹 (𝑗) Δ𝑥(𝑘) + 𝑦𝑖 (𝑘) ≤ 𝑖 𝑖 𝑦𝑚𝑎𝑥𝑗𝑖 , 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛𝑦 ; 𝑘 ≥ 0, 𝑗 = 1, 2 ⋅ ⋅ ⋅ , 𝑁𝑦 , then inequalities (32) are equivalent to: ( ) ( ) 𝑦𝑚𝑖𝑛𝑗𝑖 − 𝐹 (𝑗) Δ𝑥(𝑘) − 𝑦𝑖 (𝑘) ≤ 𝐻 (𝑗) Δu ≤ 𝑖 ( 𝑖 ) 𝑦𝑚𝑎𝑥𝑗𝑖 − 𝐹 (𝑗) Δ𝑥(𝑘) − 𝑦𝑖 (𝑘), 𝑖 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛𝑦 ; 𝑘 ≥ 0, 𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝑁𝑦 .

(32)

(33)

The notation 𝐻 (𝑗) (or 𝐹 (𝑗) ) means that one must take the 𝑛𝑦 rows of 𝐻 (or 𝐹 ) corresponding to each output prediction step 𝑦ˆ(𝑘 + 𝑗∣𝑘) in y ˆ (22). In inequalities (33), both sides are LMIs in Δu. So, there are 2𝑛𝑦 𝑁𝑦 LMIs in (33). 4. THE CONSTRAINED STABLE MPC In this section, the stability of the closed loop system is analyzed. This result is attained by imposing a nonincreasing monotonicity future behavior for the upper bound of the cost function. A sufficient condition is obtained in LMI form that guarantees a stable control law, when added to MPC problem (21). Hence, the MPC problem, subject to input and output constraints, is reduced to a convex optimization involving LMI. The idea used in this formulation, that is, to relate closed loop stability to monotonicity of the cost function, has been exploited in Scokaert and Clarke [1994] , Zheng and Morari [1993], Oliveira et al. [2000], but using others methodologies. Firstly, an upper bound 𝛾 > 0 is considered on performance objective (21): 𝐽(𝑘) ≤ 𝛾. From (23), 𝐽(𝑘) ≤ 𝛾 can be rewritten as 𝑇

[𝐻Δu + 𝐹 Δ𝑥(𝑘) + Π𝑦(𝑘) − r] 𝑄𝑦 (34) [𝐻Δu + 𝐹 Δ𝑥(𝑘) + Π𝑦(𝑘) − r] + Δu𝑇 𝑅Δu ≤ 𝛾. By using Schur complement, this upper bound satisfies inequality ⎤ ⎡ 1 𝛾

⎣∗ ∗

[𝐻Δu + 𝐹 Δ𝑥(𝑘) + Π𝑦(𝑘) − r]𝑇 𝑄𝑦2 𝐼 ∗

1

Δu𝑇 𝑅 2 0 𝐼

⎦ ≥ 0, (35)

The symbol ∗ is used throughout this paper to induce a symmetric structure. As Δ𝑥(𝑘), r and 𝑦(𝑘) are available at each sampling time 𝑘, inequality (35) is LMI in Δu and 𝛾. Define the matrices 𝐹¯ ∈ 𝑅(𝑛𝑦 𝑁𝑦 )×𝑛 and ¯ r as ⎡ ⎤ ⎡ ⎤ 𝐶𝑆1 𝐴 𝑟(𝑘 + 2) ⎢ 𝐶𝑆2 𝐴 ⎥ ⎢ 𝑟(𝑘 + 3) ⎥ ⎥ and ¯ ⎢ ⎥, 𝐹¯ = ⎢ r = . .. ⎣ ⎦ ⎣ ⎦ .. . 𝐶𝑆𝑁𝑦 𝐴

𝑟(𝑘 + 𝑁𝑦 + 1)

¯ ∈ 𝑅(𝑛𝑦 𝑁𝑦 )×[𝑛𝑢 (𝑁𝑢 +1)] and 𝑇 and matrix 𝐻 ⎡ ⎤ 𝐶𝑆1 𝐵 𝐶𝑆0 𝐵 0 ⋅⋅⋅ 𝐶𝑆1 𝐵 𝐶𝑆0 𝐵 ⋅ ⋅ ⋅ ⎥ ⎢ 𝐶𝑆2 𝐵 ¯ =⎢ ⎥, 𝐻 .. .. .. ⎣ ⎦ . . . 𝐶𝑆𝑁𝑦 𝐵 𝐶𝑆𝑁𝑦 −1 𝐵 𝐶𝑆𝑁𝑦 −2 ⋅ ⋅ ⋅ (36) 𝑇 = [ 𝐼 0 0 ⋅ ⋅ ⋅ 0 ] ∈ 𝑅𝑛𝑢 ×(𝑛𝑢 𝑁𝑢 ) , where the identity matrix 𝐼 ∈ 𝑅𝑛𝑢 ×𝑛𝑢 . ¯ (36) can be partitioned as Matrices 𝐻 [ ] ¯ 𝐻 , ¯ = ℎ 𝐻

(37)

¯ ∈ 𝑅(𝑛𝑦 𝑁𝑦 )×𝑛𝑢 and 𝐻 ∈ 𝑅(𝑛𝑦 𝑁𝑦 )×(𝑛𝑢 𝑁𝑢 ) . where ℎ Before discussing the main result of this section, additional constraints on inputs and outputs at the sample time (𝑘 + 1) are considered. Thus defining the control vector ¯ the for component-wise at sample time (𝑘 + 1) as Δu, peak bounds on input (25) at (𝑘 + 1) are given by ¯ 𝑢𝑚𝑖𝑛0𝑖 − (𝑇 )𝑖 Δu − 𝑢𝑖 (𝑘 − 1) ≤ ([ 𝐼 0 0 ⋅ ⋅ ⋅ 0 ])𝑖 Δu ≤ 𝑢𝑚𝑎𝑥0𝑖 − (𝑇 )𝑖 Δu − 𝑢𝑖 (𝑘 − 1), ¯ 𝑢𝑚𝑖𝑛1𝑖 − (𝑇 )𝑖 Δu − 𝑢𝑖 (𝑘 − 1) ≤ ([ 𝐼 𝐼 0 ⋅ ⋅ ⋅ 0 ])𝑖 Δu (38) ≤ 𝑢𝑚𝑎𝑥1𝑖 − (𝑇 )𝑖 Δu − 𝑢𝑖 (𝑘 − 1), .. . ¯ 𝑢𝑚𝑖𝑛(𝑁𝑢 −1)𝑖 − (𝑇 )𝑖 Δu − 𝑢𝑖 (𝑘 − 1) ≤ ([ 𝐼 𝐼 𝐼 ⋅ ⋅ ⋅ 𝐼 ])𝑖 Δu ≤ 𝑢𝑚𝑎𝑥(𝑁𝑢 −1)𝑖 − (𝑇 )𝑖 Δu − 𝑢𝑖 (𝑘 − 1), with 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛𝑢 . In a similar way, constraints are defined for the outputs at sample time (𝑘 + 1), for component-wise peak bounds (27) ( ) ( ) ¯ (𝑗) 𝑇 Δu+ 𝑦𝑚𝑖𝑛𝑗𝑖 − 𝐹¯ (𝑗) Δ𝑥(𝑘) − 𝑦𝑖 (𝑘) ≤ ℎ 𝑖 𝑖 ) ) ( ( ¯ ≤ 𝑦𝑚𝑎𝑥𝑗𝑖 − 𝐹¯ (𝑗) Δ𝑥(𝑘) − 𝑦𝑖 (𝑘), (39) 𝐻 (𝑗) Δu 𝑖 𝑖 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛𝑦 ; 𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝑁𝑦 . Theorem 1: Consider discrete-time system (19) and let 𝑦(𝑘) and Δ𝑥(𝑘) be respectively the measured output and the calculated state variation, at sample 𝑘. A MPC problem with guaranteed stability, with input and output constraints, can be solved by the following convex optimization problem: min 𝛾𝑓 (40) ¯ Δu,Δu,𝛾>0,𝛾 𝑓 >0

subject to ⎡ ⎤ 1 1 𝑇 𝛾 [𝐻Δu + 𝐹 Δ𝑥(𝑘) + Π𝑦(𝑘) − r] 𝑄𝑦2 Δu𝑇 𝑅 2 ⎣∗ ⎦ ≥ 0, 𝐼 0 ∗ ∗ 𝐼

⎡

𝛾𝑓 ⎣ ∗ ∗

⎤

¯ 𝑇𝑅 𝑁 (𝑘) Δu ⎦ ≥ 0, 𝐼 0 ∗ 𝐼 1 2

(41) (42) 1

¯ Δu + 𝐻 Δu ¯ + 𝐹¯ Δ𝑥(𝑘)+ Π𝑦(𝑘) − ¯ with 𝑁 (𝑘) = [ℎ𝑇 r]𝑇 𝑄𝑦2 , 𝛾𝑝 ≥ 𝛾,

(43)

𝛾 ≥ 𝛾𝑓 ,

(44)

𝛾 ≥ 𝛾𝜖 ,

(45)

and constraints (30), (31), (38), (33) and (39), depending on the input and output constrains to be imposed, respectively. 𝛾𝑝 is the optimal value of 𝛾 computed at sample time (𝑘 − 1). 𝛾𝜖 is a tuning parameter. At each sampling time 𝑘, two sequences of future control moves are calculated, ¯ at time (𝑘 + 1). Only the Δu starting at time 𝑘 and Δu, first control move of Δu is applied to the system at time 𝑘. This receding horizon control law guarantees that the upper bound of the cost function is monotonically nonincreasing, leading to the BIBO stability of closed loop system (19). Proof: LMIs (35), presented previously, and (42) are related to the cost upper-bound. LMI (42) is similar to (35) for Δ¯u . LMIs (42), (43) and (44) guarantee the stability of the closed loop system, as shown in the following. The non-increasing monotonicity for the upper bound is obtained if 𝛾𝑝 ≥ 𝛾 (∗) ≥ 𝛾𝑓 (∗) , ∀𝑘 ≥ 0,

(46)

where (∗) means the optimal value calculated by problem (40) at each sample time. LMI (43) guarantees that the left part of inequality (46) is true, and 𝛾 (∗) ≥ 𝐽 (∗) (𝑘), from LMI (41). The right-hand-side of inequality (46) is (∗) achieved by means of LMI (44), and 𝛾𝑓 ≥ 𝐽 (∗) (𝑘 + 1), from LMI (42). The condition obtained from LMI (42) is approached as follows. Using Schur complement, LMI (42) can be obtained from the fact of vector y ˆ calculated at sample time (𝑘 + 1) is given by ¯ Δu + 𝐻 ¯ + 𝐹¯ Δ𝑥(𝑘) + Π𝑦(𝑘). ¯ Δu y ˆ = ℎ𝑇 (47) (∗)

So, LMI (42) guarantees that 𝛾𝑓 ≥ 𝐽 (∗) (𝑘 + 1). As the upper-bound is monotonically non-increasing, to avoid premature stop of the algorithm when great changes in reference input happen, a slack variable 𝛾𝜖 is included. LMIs (30), (31), (38), (33) and (39) assure that the inputs and outputs constraints at time 𝑘 and (𝑘 + 1) be satisfied, and the proof is complete. □ Remark: Constraints on the variation of the control inputs ¯ are not considered. at 𝑘 + 1, i.e., Δu, Thus, Theorem 1 contains a new finite horizon predictive controller that, for the constant set-point case, guarantees the monotonicity of the cost function upper bound, so the cost function does not increase through time implying closed loop stability.

5. TRAJECTORY GENERATION

with contraints on ∆u without contraints on ∆u

Motor Voltages 5

u1

The trajectory of reference is defined as a set of points in the world frame (𝑂𝑋𝑌 ), see Figure 2: 𝑇 𝑟𝑎𝑗(𝑘 + 𝑗) = ¯ + 𝑗)]𝑇 , 𝑗 = 0, 1, ..., 𝑁𝑦 − 1. So, [𝑥¯𝑟 (𝑘 + 𝑗) 𝑦¯𝑟 (𝑘 + 𝑗) 𝜃(𝑘 given the current position and heading of the robot, it is necessary to calculate the desired velocities for the next 𝑁𝑦 periods of time. The vector of velocity references 𝑥 ¯(𝑘 + 𝑇 𝑗∣𝑘)=[¯ 𝑣 (𝑘+𝑗∣𝑘) 𝑣¯𝑛 (𝑘+𝑗∣𝑘) 𝑤(𝑘+𝑗∣𝑘)] ¯ , 𝑗 = 0, 1, ..., 𝐻𝑝− 1, where 𝑗 is an step prediction of the robot velocity made at instant 𝑘, is given by

0

−5 0

10

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60

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90

0

10

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90

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90

u2

5

0

−5

u3

5

0

−5

Time [samples]

Fig. 3. Control signals. with contraints on ∆u without contraints on ∆u

v(m/s)

1

0.5

0

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vn(m/s)

1 0.5 0 −0.5 4

[

𝑣¯(𝑘 + 𝑗∣𝑘) 𝑣¯𝑛 (𝑘 + 𝑗∣𝑘) 𝑤(𝑘 ¯ + 𝑗∣𝑘)

]

[ =

𝑐𝑜𝑠(𝜃(𝑘)) 𝑠𝑖𝑛(𝜃(𝑘)) 0 −𝑠𝑖𝑛(𝜃(𝑘)) 𝑐𝑜𝑠(𝜃(𝑘)) 0 0 0 1

][

𝑒𝑣𝑥 𝑒𝑣𝑦 𝑒𝑤

w(rad/s)

Fig. 2. Trajectory of reference. ]

0 −2

, (48)

with

2

Time [samples]

Fig. 4. Output signals. [

𝑒𝑣𝑥 𝑒𝑣𝑦 𝑒𝑤

]

[ =

𝑣𝑛𝑎𝑣 𝑐𝑜𝑠(𝜑) 𝑣𝑛𝑎𝑣 𝑠𝑖𝑛(𝜑) ¯ + 𝑗∣𝑘) − 𝜃(𝑘) 𝜃(𝑘

] ,

(49)

reference with contraints on ∆u without contraints on ∆u

1.2

1

and

𝜉(𝑘) = [𝑥𝑟 (𝑘) 𝑦𝑟 (𝑘) 𝜃(𝑘)]𝑇 is the robot pose at time step 𝑘, and 𝑣𝑛𝑎𝑣 is the linear velocity of navigation of the robot, which is a design parameter.

Y [meters]

0.8

𝜑 = 𝑎𝑡𝑎𝑛2 (𝑦¯𝑟 (𝑘 + 𝑗∣𝑘) − 𝑦𝑟 (𝑘), 𝑥¯𝑟 (𝑘 + 𝑗∣𝑘) − 𝑥𝑟 (𝑘))(50) .

0.6

0.4

0.2

6. SIMULATION RESULTS 0

In this section, the MPC with output and input constraints is used to control the omni-directional robot with 3 wheels. The controller parameters are presented in the following. The cost function weight matrices are defined as 𝑄𝑦 = 100𝐼 and 𝑅 = 0.01𝐼, and the slack variable is 𝛾𝜖 = 800. The prediction and control horizons are selected as 𝑁𝑦 = 5 and 𝑁𝑢 = 3. The input and output signals constraints are ∣𝑢𝑖 ∣ ≤ 6, 𝑖 = 1, 2, 3, ∣Δ𝑢𝑖 (𝑘 + 𝑗∣𝑘)∣ ≤ 1, 𝑖 = 1, 2, 3, 𝑗 ≥ 1, and ∣𝑦𝑖 ∣ ≤ 2.02, 𝑖 = 1, 2, ∣𝑦3 ∣ ≤ 20.02. The linear velocity of navigation was 𝑣𝑛𝑎𝑣 = 0.6(𝑚/𝑠). The trajectory of test has sudden change of direction and orientation with movements of rotation and translation in the same time, in order to test the controller in hard condition. On the first part of the trajectory, the robot orientation is fixed in 0 deg. On the second part 90 deg, on the third part 180 deg, and on the last part the robot

−0.2 0.2

0.3

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1

1.1

1.2

X [meters]

Fig. 5. Tracking performance. is commanded to rotate 270 deg. The tracking response is illustrated in Figure 5, with good performance, the peak tracking error is under 3cm bound. Figure 7 shows the robot pose trajectory, and the orientation angle tracking error is less than 8 deg. In this test, the motor voltages does not exceed the predefined limits, as can be seen in Figure 3. The use of Δ𝑢𝑖 ≤ 1 volt represents a smooth control inputs. In the contrary case, voltage variations bigger than 6 volts can be seen, for example on the sample 70 of the Figure 3 (𝑢3 with variations from -3.1 to 3.2 volts and 𝑢1 with variations from 2 to -4.2 volts), forcing the

The output signals can be observed in Figure 4, and the upper bound for the cost function during the simulation time is presented in Figure 6. It can be point out that the upper bound for the cost function is monotonically non-increasing, as expected, since it is guaranteed by the control law. The closed loop is stable and the transitory behavior is adequate. The output signals belong to the feasible set, since they were taken into account by the control law design.

7

2.5

x 10

with contraints on ∆u without contraints on ∆u

Upper bound gamma

2

1.5

1

7. CONCLUSIONS 0.5

0

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Time [samples]

Fig. 6. Upper bound for the cost function associated to the control law.

1.2

1

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0.8

0.6

REFERENCES

0.4

0.2

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−0.2

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In this paper, it has been presented simulation results obtained from application of a new model predictive controller design, with cost function developed over finite horizon and based on LMI framework, for omni-directional mobile robots. This LMI formulation is achieved by defining an upper bound to a quadratic objective function, considering input and output constraints. It has been shown that the control method is adequate for controlling robotic systems. Closed loop system stability is guaranteed by deriving LMI constraints for the monotonicity of the upper bound of the cost function. The constraints on the variation of control inputs has influence in the pathfollowing errors, and also in the time to goal. As seen in Figures, it may be concluded that constraints preserves the actuators with a reasonably performance of control.

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Fig. 7. Robot poses. actuators to operate in peak regions. Another interesting characteristic of smooth inputs was the time to goal. The robot covered a bigger distance with constraints on Δ𝑢𝑖 due to a smooth navigation, compare Figures 7(a) and 7(b).

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